Let $\varphi:(R,m) \to (S,n)$ be a local homomorphism between local noetherian rings.

Recall that the fiber of $\varphi$ is by definition the ring $F(\varphi) = S/mS$.

If $\varphi$ is flat, it is well known that the Krull dimension of $F(\varphi)$ satisfies 
$ \dim F(\varphi) = \dim S - \dim R$.

Of course, if $\varphi$ is not flat, this fails. In my case, all I know about $\varphi$ is that it is of finite flat dimension.

I wonder if there is still a formula for $ \dim F(\varphi) $ in terms of $\dim S$, $\dim R$ and some other homological invariants related to $R,S, \varphi$?